Walk-power chromatic-number conjecture for projective cubes

Let $r$ and $k$ be integers with $r\geq k$. For a graph $G$ and a positive integer $l$, let $G^{(l)}$ be the graph on the same vertex set in which two vertices are adjacent when they are joined by a walk of length $l$ in $G$. Here $\chi$ denotes chromatic number. Walk-power chromatic-number conjecture.

$$\chi\left(\mathcal{PC}_{2r}^{(2k-1)}\right)\geq 2^{2k}.$$

This conjecture would imply the surjectivity conjecture for homomorphisms between projective cubes, because $\mathcal{PC}_{2k}^{(2k-1)}$ is isomorphic to $K_{2^{2k}}$. It remains open in the paper.